Geometry-Aware FEM in Computational Mechanics

The interdisciplinary research area of computational mechanics combines concepts from computer science, applied mathematics, and engineering for the efficient simulation of mechanical structures. Applications of such simulations can be found in a wide range of areas, including engineering, biomechanics, or life sciences. In the case of real-life-applications, particular emphasis has to be put on the discrete representation of the mechanical structures under consideration. On the one hand, the possibly highly complex surfaces have to be represented in a sufficiently accurate way; on the other hand, a volume mesh of high quality is required in order to ensure the quality of the finite element approximation and the good convergence of iterative solution methods.

Although state-of-the-art mesh generation methods and tools allow for a relatively comfortable generation of meshes, the creation of a high quality mesh for a complex structure still requires considerable effort and is a time-consuming task. Thus, it is desirable to make use of a created volume mesh as long as possible throughout the course of a simulation. However, this leads to two difficulties. The first is connected to surface representation and adaptivity. Using adaptive refinement strategies, a higher resolution at the boundary should be accompanied by a better approximation of the real surface of the structure under consideration. This requires changing the mesh along the boundary, which in turn may have a considerable influence on the quality of the volume mesh. The second is related to the degradation of mesh quality during time-dependent simulations which involve large deformations. Despite the fact that in case of a highly deformed mesh a complete remeshing will be necessary, the quality of the mesh might decrease gradually during the simulation process, thus affecting the quality of the simulation results and the speed of the simulation process significantly.

The main idea of this project is to overcome these difficulties by developing and implementing a geometry-aware simulation environment for computational mechanics, which combines in a modular and interactive fashion the handling of complex geometries and volume meshes with the discretization and solution process. Thus, instead of seeing geometry approximation as a burden which has to be taken into account when computing the finite element approximations, we aim at exploiting concepts and methods from the field of geometry processing in order to guarantee a constantly high quality of the boundary approximation and the volume mesh throughout the course of a transient simulation.